Practice and Theory of Automated Timetabling (PATAT 2012), 29-31 August 2012, Son, Norway
HySST: Hyper-heuristic Search Strategies and
Timetabling
Ahmed Kheiri · Ender Özcan ·
Andrew J. Parkes
Received: date / Accepted: date
1 Introduction
High school timetabling (HST) is a well-known real-world combinatorial optimisation problem. It requires the scheduling of events and resources, such as
courses, classes of students, teachers, rooms and more within a fixed number of
time slots subject to a set of constraints. In a standard fashion, constraints are
separated into ‘hard’1 and soft. The hard constraints must be satisfied in order
to achieve feasibility, whereas the soft constraints represent preferences and a
solution for a given problem; solutions are expected to satisfy all hard constraints and as many soft constraints as possible. The HST problem is known
to be NP-complete [2] even in simplified forms. For a recent survey of HST
see [4]. Also, see [5] for a description of the specific HST version studied here,
and also of the third international timetabling competition, ITC2011. Briefly,
the ITC2011 problem instances contain 15 types of constraints and a candidate
solution is evaluated in terms of two components: feasibility and preferences.
The evaluation function computes the weighted hard and soft constraint violations for a given solution as infeasibility and objective values, respectively.
For the comparison of algorithms, a solution is considered to be better than
another if it has a smaller infeasibility value, or an equal infeasibility value
and a smaller objective value.
Our approach to solving the HST is based on development of a hyperheuristic (see [1] for a recent survey) in order to intelligently control the
application of a range of neighbourhood move operators. Hyper-heuristics
explore the space of (meta-)heuristics, as opposed to directly searching the
space of solutions, and generally, split into one of two types: selection hyperheuristics that select between existing low-level heuristics, and generation
University of Nottingham, School of Computer Science
Jubilee Campus, Wollaton Road, Nottingham, NG8 1BB, UK
{ axk, exo, ajp }@cs.nott.ac.uk
1 In the competition, such constraints are not strictly hard but are simply much more
heavily penalised than the ’soft’ constraints.
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Practice and Theory of Automated Timetabling (PATAT 2012), 29-31 August 2012, Son, Norway
Algorithm HySST Solver ITC2011
S = create initial solution(); // takes tinit time
tremaining = toverall − tinit
Sbest = S;
while (tremaining notExceeded)
Sstage best = S;
Sstage start = S;
while (tstage notExceeded) // stage entry
LLH = SelectRandomlyF rom(M utationalHeuristics);
S ′ = ApplyHeuristic(LLH, S);
if (S ′ isBetterT han Sbest ) then Sbest = S ′ ;
if (S ′ isBetterT han Sstage best ) then Sstage best = S ′ ;
S = M oveAcceptance(S, S ′ , ϵ); // can accept with factor (1 + ϵ) worse
if (Sstage best isN otBetterT han Sstage start ) then
S = ApplyHillClimbing(S); // uses the hill climbing heuristics
return Sbest ;
Fig. 1 Pseudocode of the implemented hyper-heuristic. Here ϵ is a small (tunable) parameter that controls the level of worsening moves that are acceptable.
hyper-heuristics that build the decision procedures used with the heuristics. A
selection hyper-heuristic generally includes both heuristic selection and move
acceptance within a single point search framework (that is, without the use of
populations of solutions): At each iteration, a candidate new solution is produced by selecting and applying a heuristic (neighbourhood operator) from
a set of low level heuristics; A ‘move acceptance’ component then decides
whether or not the candidate should replace the incumbent solution.
In this work, we develop and exploit a generalised selective hyper-heuristic
(though envisage that future work could well exploit generative methods). We
build on a previous study [3] that demonstrated the effectiveness of a generalised version of the iterated local search approach. Specifically, our hyperheuristic uses a structured and staged application of multiple perturbative and
hill climbing operators as opposed to simple selection from a single pool of all
operators.
2 Hyper-heuristic Search for High School Timetabling
The search algorithm is implemented as a time contract algorithm and completes its execution in toverall time. It consists of an initial construction phase
followed by an extended improvement; see Figure 1 for the pseudocode. The
initial construction of a complete solution is performed using the general solver
implemented by Jeff Kingston in the KHE library2 . The improvement phase
uses the remaining time left (tremaining ) after the construction of the initial solution which takes tinit time. Subsequently, a hyper-heuristic is used to control
and mix a set of 11 low-level domain-specific heuristics and that are (mostly)
2
http://sydney.edu.au/engineering/it/~jeff/khe/
Practice and Theory of Automated Timetabling (PATAT 2012), 29-31 August 2012, Son, Norway
fairly simple moves such as moving a task to a different resource, or swaps of
events. They are divided into two sets; 9 mutational operators that do a randomised move, and 2 hill climbing operators that search their neighbourhoods
for better solutions. Note that the construction phase often gives a solution in
which hard constraints are violated, and so the improvement phase also needs
to improve the hard constraints. The hyper-heuristic divides the search into
stages where each stage takes a prefixed amount of time (tstage ). In each stage,
firstly and repeatedly, one of the 9 mutational heuristics is applied and the
move is accepted if it is improving or is not worsening by more than a small
amount. If no improvement is achieved during a stage, a hill climbing phase is
applied using the 2 hill climbing heuristics. Unlike most of the mutational operators, these 2 hill-climbing heuristics are capable of making quite large changes
to a solution. The hill climbing phase is itself also slightly non-standard. One
of the operators is designed using neighbourhood structures based on ejection
chains while the other operator is a type of first improvement hill climbing
operator. Both hill climbing operators attempt to make moves which respect
a particular constraint type while hoping to improve upon the other types of
constraint violations but might have a net worsening of the objective, however, then such worsening moves are rejected. A hill climbing step is always
non-worsening and so can be repeatedly applied in standard fashion until a
local minimum is reached. A notable difference from standard methods (such
as in memetic algorithms) is that we found that performance is better if the
hill climbing is not applied if the mutational operators managed to improve
the best solution. We suspect that excessive use of the hill climbing somehow
gives over-optimised local solutions that afterwards lead to restricted movement within the search space.
In summary, we have developed and applied a hyper-heuristic to intelligently and effectively exploit a suite of neighbourhood move operators. Since
the aim of hyper-heuristics is to separate adaptive search control from the
details of the specific domain, we naturally also envisage our hyper-heuristic
could be applied to other PATAT-related problems, such as to the timetabling
of university examinations and courses.
References
1. Burke, E.K., Gendreau, M., Hyde, M., Kendall, G., Ochoa, G., Özcan, E., Qu, R.: Hyperheuristics: A survey of the state of the art. Journal of the Operational Research Society
(to appear)
2. Even, S., Itai, A., Shamir, A.: On the Complexity of Timetable and Multicommodity
Flow Problems. SIAM Journal on Computing 5(4), 691–703 (1976)
3. Özcan, E., Bilgin, B., Korkmaz, E.E.: A comprehensive analysis of hyper-heuristics. Intelligent Data Analysis 12(1), 3–23 (2008)
4. Pillay, N.: An overview of school timetabling research. In: Proceedings of the 8th International Conference on the Practice and Theory of Automated Timetabling (PATAT
2010), pp. 321–335 (2010)
5. Post, G., Gaspero, L.D., Kingston, J.H., McCollum, B., Schaerf, A.: The third international timetabling competition. In: Proceedings of the Ninth International Conference
on the Practice and Theory of Automated Timetabling (PATAT 2012) (2012)
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